bibkey: dinov2026kime authors: Ivo D. Dinov year: 2026 title: “Kime-Representation Formulations of Three Open Problems in the Foundations of Classical Mechanics: Uncertainty, Invariant Entropy, and Directional Degrees of Freedom” doi: 10.48550/arxiv.2607.07851 url: https://arxiv.org/abs/2607.07851v1 claim: “Conjecture 3.22(ii) asserts that the phase-equipartitioned product state minimizes every within-DOF uncertainty at a fixed action marginal and entropy lower bound.” strata_touched:

• D5/S3/Quantum/Information/DinovOriginalMinimumRefutation license: citation-only triage: anchor

Equipartition duality and the within-DOF minimum assertion

The source is arXiv:2607.07851v1, Conjecture 3.22 (Equipartition duality):

Fix $n\ge 2$, an entropy value $s$, and an action marginal ${\rho }_{\mathbf{J}}$ on $(0,\infty {)}^n$. Among all states on ${\mathbb{T}}^n\times (0,\infty {)}^n$ with entropy $\ge s$ and action marginal ${\rho }_{\mathbf{J}}$, the phase-equipartitioned product state (unique when it exists) simultaneously (i) maximizes the entropy, (ii) minimizes every within-DOF uncertainty $u_j$, and (iii) is the unique state at which the per-DOF conjectured bound of Problem 3.17(b) is saturated for all $j$; moreover it is the unique fixed point, with the given marginal, of the multi-DOF kime-deformed semigroup ${\partial }_t\stackrel{=\sum j(-\omega j\partial \theta j+\epsilon \partial \theta j2)\rho }{\rho }$.

Definition 2.4 uses probability densities. Lemma 2.3 gives the source map $q=\sqrt{2J}\sin \theta$, $p=\sqrt{2J}\cos \theta$, with angular measure of mass $2\pi$ and Lebesgue measure on positive actions. The uncertainty in Theorem 3.16 is the square root of the determinant of the actual two by two coordinate covariance block. The comparison uses an entropy lower bound, not entropy equality, and imposes no fixed symplectic orbit.

The repository refutation uses $n=2$, $s=0$ and $r(J_1,J_2)=e^{-(J_1+J_2)}$. The independent nonuniform phases have density $(1+\frac{1}{2}\cos 2\theta )/(2\pi )$; their uniform-phase comparator has the same normalized joint action law. Their actual Cartesian covariances are $\mathrm{diag}(3/4,5/4,3/4,5/4)$ and $I_4$, respectively. Both within-DOF uncertainties are $\sqrt{15/16}<1$ in the nonuniform state. Both states have finite nonnegative entropy, L2 coordinates and positive definite full covariance. Source pushforwards, angular fibers and entropy transport are proved inside the single result.

The original all-$n$ assertion implies clause (ii), and clause (ii) implies the certified $n=2$ predicate. The formal result negates this necessary predicate; it does not formulate a generic-$n$ transport theorem or separately settle entropy maximization, saturation uniqueness or the semigroup claims. The Cartesian density at zero action is a Borel zero extension on null planes. The dynamical smoothness and decay hypotheses of neighboring Theorem 3.21 are not added to the static conjecture.

Literature boundary

The accepted independent source reviews report the following bounded readings: arXiv history contains only v1; the kime/equipartition topic query has one result, this paper; OpenAlex work W7167927072 has zero indexed citations; MathDB target 375895 has no posted solution and an unrefreshed progress panel. Semantic Scholar returned HTTP 429. No proof or refutation was found in the successfully checked scopes. This does not establish exhaustive literature absence or worldwide priority. The readings and Tier 1 classification are attached to preregistration #13139.

Verified locator

• DOI: https://doi.org/10.48550/arxiv.2607.07851
• URL: https://arxiv.org/abs/2607.07851v1
• Original HTML: https://arxiv.org/html/2607.07851v1#S3.Thmtheorem22

The source HTML has SHA-256 485c13cf847cd636050a971a324b4c2943dcbdfd76a913c7706ce5ff35817095. Only citation and the necessary source assertion are retained; the source’s CC BY-NC-SA 4.0 license is not assigned to the repository’s original proof.